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Fixing the Haskell syntax
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Pseudonym ♦
Pseudonym ♦
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So I'm almost done teaching myself category theory. One of the main take-aways for me is that type constructors (higher-kinded types) are endo-functors.

But it this always the case? What's throwing me off is Haskell's Functor typeclass:

class Functor f :: * -> * where
   fmap :: (a -> b) -> (f a -> f b)

If all type constructors were a priori functors, there would be no need to define a special type-class. But I can't think of any type-constructors that fail to satisfy the Functor type-class: product, sum, function, list, set, option, etc.

I also cannot imagine the utility of a type-constructors that is "one-half" of a functor, i.e. it has a map from objects to objects, but no map between morphisms.

So I'm almost done teaching myself category theory. One of the main take-aways for me is that type constructors (higher-kinded types) are endo-functors.

But it this always the case? What's throwing me off is Haskell's Functor typeclass:

class Functor f: * -> * where
   fmap: (a -> b) -> (f a -> f b)

If all type constructors were a priori functors, there would be no need to define a special type-class. But I can't think of any type-constructors that fail to satisfy the Functor type-class: product, sum, function, list, set, option, etc.

I also cannot imagine the utility of a type-constructors that is "one-half" of a functor, i.e. it has a map from objects to objects, but no map between morphisms.

So I'm almost done teaching myself category theory. One of the main take-aways for me is that type constructors (higher-kinded types) are endo-functors.

But it this always the case? What's throwing me off is Haskell's Functor typeclass:

class Functor f :: * -> * where
   fmap :: (a -> b) -> (f a -> f b)

If all type constructors were a priori functors, there would be no need to define a special type-class. But I can't think of any type-constructors that fail to satisfy the Functor type-class: product, sum, function, list, set, option, etc.

I also cannot imagine the utility of a type-constructors that is "one-half" of a functor, i.e. it has a map from objects to objects, but no map between morphisms.

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gardenhead
gardenhead
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Are there any type constructors which are *not* functors?

So I'm almost done teaching myself category theory. One of the main take-aways for me is that type constructors (higher-kinded types) are endo-functors.

But it this always the case? What's throwing me off is Haskell's Functor typeclass:

class Functor f: * -> * where
   fmap: (a -> b) -> (f a -> f b)

If all type constructors were a priori functors, there would be no need to define a special type-class. But I can't think of any type-constructors that fail to satisfy the Functor type-class: product, sum, function, list, set, option, etc.

I also cannot imagine the utility of a type-constructors that is "one-half" of a functor, i.e. it has a map from objects to objects, but no map between morphisms.